Extreme value theory for a class of nonstationary time series with applications

Abstract

Consider a class of nonstationary time series with the form $Y_t = \mu_t + \xi_t$ where ${\xi_t}$ is a sequence of infinite moving averages of independent random variables with regularly varying tail probabilities and different scale parameters. In this article, the extreme value theory of ${Y_t}$ is studied. Under mild conditions, convergence results for a point process based on the moving averages are proved, and extremal properties of the nonstationary time series, including the convergence of maxima to extremal processes and the limit point process of exceedances, are derived. The results are applied to the analysis of tropospheric ozone data in the Chicago area. Probabilities of monthly maximum ozone concentrations exceeding some specific levels are estimated, and the mean rate of exceedances of daily maximum ozone over the national standard 120 ppb is also assessed.